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Unformatted text preview: 11. Appendix 1: A Very Brief Linear ALgebra Review Introduction. Linear Algebra, also known as matrix theory, is an important ele- ment of all branches of mathematics. Very often in this course we study the shapes and the symmetries of molecules. Motion of 3D space which leave molecules rigid can be described by matrices. Briefly mentioned in these notes will be quantum mechanics, where matrices and their eigenvalues have an essential role. In all cases it is useful to allow the entries in the matrix to be complex numbers. If you have studied matrices only with real number entries, it is very easy to adapt to complex numbers. Almost all the rules of computations are the same. In doing computations with matrices it is useful to have a computer program such as Maple or Matlab . These tools make multiplication of matrices very easy, and they work with complex numbers. The main difference between Maple and Matlab is that Maple can work symbolically, that is, you can use letter as well as numbers for entries. When using numbers, Matlab is often faster. Below we give a review of a few basic ideas that will be used in the course. Matrices. Example : A = 2 1- 1 3 A is a matrix with 2 rows and 2 columns i.e a 2 × 2 matrix. A matrix with m rows and n columns is called an m × n matrix. A matrix with the same number of rows and columns is called a square matrix. 3 × 3 square matrix: B = 3 1 7- 1 2 0 1 5 3 × 2 matrix: C = 2- 9 10 1 14 A 1 × 1 matrix is the same as a number or scalar, 3 =  . Vectors. Matrices with 1 row are called row vectors and matrices with 1 column are called column vectors. A = 2 1 B = 3 2 1 are column vectors. C = 2 1 D = 3 2 1 are row vectors. Usually we will assume vectors are column vectors. A row vector can be converted into a column vector (or vice versa) by the transpose operation, which changes rows to columns. 1 2 Example : 1- 1 0 t = 1- 1 Complex Numbers. Complex numbers can be used in matrices....
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This note was uploaded on 01/15/2012 for the course MAP 5485 taught by Professor Staff during the Fall '11 term at FSU.
- Fall '11